Vertex Bounds in Triangulated d-Manifolds and an Application to 4-Manifold Complexity

We investigate face numbers of generalised triangulations of manifolds in arbitrary dimensions. This is motivated by the study of connections between the combinatorics of triangulations and topological properties of their underlying manifolds. For an $n$-facet triangulation of an odd-dimensional $d$-manifold with $n \geq d$, we prove that the number of vertices satisfies $v \leq n + \frac{d - 1}{2}$. Moreover, we show that this bound is tight for all odd $d$ and all $n \geq d$. For even dimensions, we conjecture the bound $v \leq \frac{n}{2} + d$. We prove that, if true, the bound is tight. Although we cannot prove the conjecture for arbitrary generalised triangulations, we prove it in the case of balanced triangulations, and provide a sufficient condition on the dual graph of the triangulation for it to hold. Furthermore, we construct families of $d$-dimensional triangulations with singularities that exceed the bound $v > \frac{n}{2} + d$, thereby demonstrating that the manifold condition is necessary for the validity of the conjecture. Our study is further motivated by an application to triangulations of simply connected $4$-manifolds: For $d=4$, our conjecture implies that a triangulation $\mathcal{T}$ of a simply connected 4-manifold $\mathcal{M}$ with $n$ pentachora satisfies $2\beta_2(\mathcal{M}) \leq n$ as a lower bound on its Matveev complexity - a bound that is known to be best possible up to a very small additive constant.